On exceptional sets in the metric Poissonian pair correlations problem

Lachmann, Thomas; Technau, Niclas

doi:10.1007/s00605-018-1199-2

Cited by 15 publications

(18 citation statements)

References 24 publications

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“…Recently, the result of Bourgain has been further extended, see [1,16,17,18]. The result given in [17] is an easy consequence of our Theorem 1 stated below and will be shown in Section 2.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 88%

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Some negative results related to Poissonian pair correlation problems

Larcher

Stockinger

2020

Discrete Mathematics

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We say that a sequence (xn) n∈N in [0, 1) has Poissonian pair correlations iffor every s ≥ 0. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence (xn) n∈N of real numbers in [0, 1) having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and in more general LS-sequences of points and digital (t, 1)-sequences. Additionally, this theorem enables us to derive negative pair correlation properties for sequences of the form ({anα}) n∈N , where (an) n∈N is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of α. Second, in this note we study the pair correlation statistics for sequences of the form, xn = {b n α}, n = 1, 2, 3, . . ., with an integer b ≥ 2, where we choose α as the Stoneham number and as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property. Throughout this article {·} denotes the fractional part of a real number. arXiv:1803.05236v2 [math.NT]

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“…Recently, the result of Bourgain has been further extended, see [1,16,17,18]. The result given in [17] is an easy consequence of our Theorem 1 stated below and will be shown in Section 2.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 88%

“…We mention that due to a result of Lachmann and Technau ( [16]), we can immediately deduce that for almost all α the pair correlations of ({a n α}) n∈N are not Poissonian, if (a n ) n∈N is a quasi-arithmetic sequence of degree d ≥ 2.…”

Section: Corollarymentioning

confidence: 90%

Some negative results related to Poissonian pair correlation problems

Larcher

Stockinger

2020

Discrete Mathematics

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“…• There exists a sequence having additive energy of order N 3 log N log log N which does not have the metric pair correlation property [10].…”

Section: Introductionmentioning

confidence: 99%

“…• For every ε > 0 there exists a sequence having additive energy of order N 3 log N (log log N ) 1+ε which has the metric pair correlation property (unpublished, but not difficult to construct using methods from [4,10]).…”

Section: Introductionmentioning

confidence: 99%

There Is No Khintchine Threshold for Metric Pair Correlations

2019

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We consider sequences of the form (anα) n mod 1, where α ∈ [0, 1] and where (an) n is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all α in the sense of Lebesgue measure, we say that (an)n has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of (an)n. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterises the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence (an)n having large additive energy which, however, maintains the metric pair correlation property.

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“…Then, ({p n α}) n≥1 does not have PPC for almost all α.The region between O N log N ) C , C > 1, and Ω (N 3 ) is therefore the interesting region and one might speculate about a sharp threshold which allows to fully describe the metrical pair correlation theory in terms of the additive energy. Further constructions and examples of sequences in this "interesting region", with an even smaller additive energy compared to the primes, were given by Lachmann and Technau[17]:Theorem 12.There exists a strictly increasing sequence of positive integers (a n ) n≥1 with E (a 1 , . .…”

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confidence: 99%